A365305 a(n) is the smallest nonnegative integer such that the sum of any nine ordered terms a(k), k<=n (repetitions allowed), is unique.

0, 1, 10, 91, 500, 3119, 13818, 59174, 211135, 742330, 2464208, 7616100, 19241477, 56562573

Offset: 1

Author

Kevin O'Bryant, Aug 31 2023

Keywords

• nonn
• more

Comments

This is the greedy B_9 sequence.

Examples

			a(4) != 72 because 72+1+1+1+1+1+1+1+1+0 = 10+10+10+10+10+10+10+10+0.
		

Links

• J. Cilleruelo and J. Jimenez-Urroz, B_h[g] sequences, Mathematika (47) 2000, pp. 109-115.
• Melvyn B. Nathanson, The third positive element in the greedy B_h-set, arXiv:2310.14426 [math.NT], 2023.
• Melvyn B. Nathanson and Kevin O'Bryant, The fourth positive element in the greedy B_h-set, arXiv:2311.14021 [math.NT], 2023.
• Kevin O'Bryant, A complete annotated bibliography of work related to Sidon sequences, Electron. J. Combin., DS11, Dynamic Surveys (2004), 39 pp.

Crossrefs

Row 9 of A365515.

Cf. A025582, A051912, A365300, A365301, A365302, A365303, A365304.

Programs

Python

def GreedyBh(h, seed, stopat):
    A = [set() for _ in range(h+1)]
    A[1] = set(seed)    # A[i] will hold the i-fold sumset
    for j in range(2,h+1): # {2,...,h}
        for x in A[1]:
            A[j].update([x+y for y in A[j-1]])
    w = max(A[1])+1
    while w <= stopat:
        wgood = True
        for k in range(1,h):
            if wgood:
                for j in range(k+1,h+1):
                    if wgood and (A[j].intersection([(j-k)*w + x for x in A[k]]) != set()):
                        wgood = False
        if wgood:
            A[1].add(w)
            for k in range(2,h+1): # update A[k]
                for j in range(1,k):
                    A[k].update([(k-j)*w + x for x in A[j]])
        w += 1
        return A[1]
GreedyBh(9,[0],10000)

Python

from itertools import count, islice, combinations_with_replacement
def A365305_gen(): # generator of terms
    aset, alist = set(), []
    for k in count(0):
        bset = set()
        for d in combinations_with_replacement(alist+[k],8):
            if (m:=sum(d)+k) in aset:
                break
            bset.add(m)
        else:
            yield k
            alist.append(k)
            aset |= bset
A365305_list = list(islice(A365305_gen(),10)) # Chai Wah Wu, Sep 01 2023

Extensions

a(11)-a(14) from Chai Wah Wu, Sep 13 2023